Apparatus and method for model-based control

ABSTRACT

Various methods and systems for the parametric control of a process include representing the process with a process model used to generate future predictions of a process variable. In one embodiment, the process exhibits integrating behavior that is represented by a non-integrating process model. In another embodiment, an inverse of the model is filtered using a filter that includes a lead time constant that is selected to minimize a steady state error of the predicted process variable. In yet another embodiment, an array of output model values is revised or reindexed in response to a change in a time-varying parameter related to the process.

TECHNICAL FIELD

The present invention generally relates to parametric model-basedcontrol of a process, and certain aspects of the present invention haveapplication to the control of an integrating process.

BACKGROUND

In parametric internal model-based control (IMC), a controller may beused to control a process where the controller includes a model of theprocess. The accuracy of the model drives the performance of thecontroller. IMC is also known in the art as model predictive control(MPC) or simply as model-based control (MBC). For purposes of thisdisclosure, the term IMC will be used.

FIG. 1 shows a representation of a control system 10 that includes amodel 12 of an integrating process 14 that may be controlled with acontroller 16 by a predictive control technique that is represented byan inverted version of the model 12 (realizable model inverse) and anassociated filter. Parameters of the model are used to calculate, orpredict, a future process variable (PV), and generate a control output(CO) responsive to a given set point. But, as will be explained ingreater detail below, modeling the behavior of an integrating processthat may have one or more poles at origin leads to computationaldifficulties.

For any non-zero disturbance (e.g., input disturbance d₁ and/or outputdisturbance d₂), the predicted process variable (also known as acontrolled variable or CV) will grow without bound due to theintegrating nature of the conventional implementation of the model 12and associated inverted model. The same holds for any non-zero controloutput CO (also known as a manipulated variable) as the model predictionof the future state of PV also will grow without a bound.

The model output under the forgoing conditions is represented in thegraphs of FIGS. 2 a and 2 b, which share the same scale for the timeaxis. In FIG. 2 a, the control output CO is graphed. As illustrated, astep change to the control output CO is made at a given time. As will beappreciated, the control output CO change is the input value to theprocess model 12 and the process 14. The predicted process variable thatis output by the model 12 is graphed in FIG. 2 b.

A dead time D is a measurement of how long the process takes to start torespond to the CO change and may be defined by the amount of time thatelapses from the change in control point to when the predicted processvariable exceeds a noise threshold defined by a noise band. A timeconstant (or lag time) may be used as an indicator of how long theprocess takes to reach steady state after the dead time. For anintegrating process, the time constant value may be the amount of timethat elapses from the end of the dead time to when the slope of thepredicted process variable achieves 63% of the slope of the steady statePV. 95% of the slope of the steady state PV may be achieved after threetime constants.

As illustrated, using an integrating process model, the value for thepredicted process variable will grow without bound. A maximum slope ofthe predicted process variable curve (or PV maximum slope) may bedetermined and the gain (k) may be defined by the PV maximum slopedivided by the CO change. The unbounded increase in predicted processvariable is not necessarily a design issue for the control system 10,but represents a computational and numerical issue.

Three common solutions exist, but each have drawbacks. The firstsolution is to convert output disturbances (or modeling errors) to anequivalent input disturbance, such that disturbances and errors that areinput to the model are mathematically zeroed in the steady state. Thesecond solution is to reinitialize the state of the predicted processvariable from time to time to reduce the occurrence of a numericaloverflow. The third solution is to use an internally stable algorithmicinternal model control (AIMC) of a two degree of freedom IMC.

While these solutions improve the behavior of the model, they allintroduce design and computational complexities. Moreover, for anintegrating process, the design of an internal model-based controller iscomplicated by the introduction of a higher order filter as will bedemonstrated below.

With continued reference to FIG. 1, conventional representations of afirst order integrating process will now be described. Under theserepresentations, the integrating process 14 may be represented byequation 1A and the model 12 of the process may be represented byequation 2.

$\begin{matrix}{{G_{p}(s)} = {\frac{k}{s}^{- {DS}}}} & \text{Eq. 1A} \\{{G_{m}(s)} = {\frac{\hat{k}}{s}^{{- \hat{D}}S}}} & \text{Eq. 2}\end{matrix}$

The steady state process gain is represented by k in equation 1A and anestimated steady state process gain is represented by {circumflex over(k)} in equation 2. Similarly, D is the process dead time and{circumflex over (D)} is an estimated dead time. In each equation, aLaplace variable is represented by S.

The controller 16 may be represented by equation 3, in which G_(m+) ⁻¹is an invertible portion of the model 12 (equation 2) and F(s) is afilter that is designed to make the controller represented by G_(c)(s)realizable. The filter specifies a desired response for the processvariable PV (referred to as a desired trajectory of PV).

G _(c)(s)=G _(m+) ⁻¹ ·F(s)  Eq. 3

A controller computation round-off error is shown in FIG. 1 as d₃, whichrepresents a rounding error. While the rounding error may be very small,the rounding error may still lead to a numerical instability issue, asdescribed below.

The system 10 of FIG. 1 may be simplified into the forms illustrated inFIGS. 3 and 4. If, in FIG. 3, a perfect model 12 is obtained for a firstorder process 14, G_(m)(s) will equal G_(p)(s), and then both G_(m)(s)and G_(p)(s) may be represented by equation 1A. If a first order filterF(s) equaling 1/(εs+1) is used, equation 3A results, where ε is a filtertime constant.

$\begin{matrix}{G_{c} = {{G_{m +}^{- 1} \cdot {F(s)}} = \frac{s}{k\left( {{ɛ\; s} + 1} \right)}}} & {{{Eq}.\mspace{14mu} 3}A}\end{matrix}$

As indicated, G_(m+) is the invertible portion of the model and is equalto k/s. In FIG. 4, G_(IMC) may be considered the combination of G_(c)and G_(m) and represents an internal model-based controller 16′. G_(IMC)may be expressed as equation 4.

$\begin{matrix}{G_{IMC} = {\frac{G_{c}}{1 - {G_{c}G_{m}}} = {\frac{\frac{s}{k\left( {{ɛ\; s} + 1} \right)}}{1 - \frac{^{- {DS}}}{{ɛ\; s} + 1}} = \frac{s}{k\left( {{ɛ\; s} + 1 - ^{- {DS}}} \right)}}}} & {{Eq}.\mspace{14mu} 4}\end{matrix}$

In equation 5A, the process variable PV is expressed using a three termequation that represents a dynamic relationship among the value at theinput to the controller 16′ (first term where SP is the set point), thevalue at the input to the process 14 following introduction of an inputdisturbance d₁ (second term) and the value at the output of the processfollowing introduction of an output disturbance(s) d₂ (third term).

$\begin{matrix}{{{PV}(s)} = {{\frac{G_{IMC}G_{p}}{1 + {G_{IMC}G_{p}}}{SP}} + {\frac{G_{p}}{1 + {G_{IMC}G_{p}}}d_{1}} + {\frac{1}{1 + {G_{IMC}G_{p}}}d_{2}}}} & {{{Eq}.\mspace{14mu} 5}A}\end{matrix}$

The second term of equation 5A corresponds to an input disturbance tothe process 14 and it is desirable for this term to be zero at steadystate. If the second term is not zero, a steady state error occurs. Byapplying a step input for the set point (or set point change) ananalysis of how PV will behave may be made. One may represent the stepinput as r, such that SP(s) equals (1/s)r and equation 6 follows fromequation 5A.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; \frac{G_{IMC}G_{p}}{1 + {G_{IMC}G_{p}}}r}}} \\{= {s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{\frac{^{- {DS}}}{\left( {{ɛ\; s} + 1 - ^{- {DS}}} \right)}}{1 + \frac{^{- {DS}}}{\left( {{ɛ\; s} + 1 - ^{- {DS}}} \right)}}r}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{^{- {DS}}}{{ɛ\; s} + 1 - ^{- {DS}} + ^{- {DS}}}r}} = r}}\end{matrix} & {{Eq}.\mspace{14mu} 6}\end{matrix}$

It may be observed that for a change in the set point SP variable, thereis a direct correlation in the process variable PV attained at steadystate. But if a step input disturbance (e.g., d₁(s) equaling d₁/s) isintroduced, a steady state error in PV results as indicated by equation7.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; \frac{G_{p}}{1 + {G_{IMC}G_{p}}}d_{1}}}} \\{= {s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{\frac{k\; ^{- {DS}}}{s}}{1 + \frac{^{- {DS}}}{\left( {{ɛ\; s} + 1 - ^{- {DS}}} \right)}}d_{1}}}} \\{= {s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{k\left( {{ɛ\; s} + 1 - ^{- {DS}}} \right)}{s\left( {{ɛ\; s} + 1 - ^{- {DS}} + ^{- {DS}}} \right)}d_{1}}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{{k\; ɛ} + {{kD}\; ^{- {DS}}}}{{ɛ\; s} + 1 + {ɛ\; s}}d_{1}}} = {{\left( {{k\; ɛ} + {k\; D}} \right)d_{1}} \neq 0}}}\end{matrix} & {{Eq}.\mspace{14mu} 7}\end{matrix}$

It may further be observed that for a step output disturbance whered₂(s) equals d₂/s, the third term of equation 5A advantageously goes tozero as demonstrated by equation 8.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; \frac{1}{1 + {G_{IMC}G_{p}}}d_{2}}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{{ɛ\; s} + 1 - ^{- {DS}}}{{ɛ\; s} + 1 - ^{- {DS}} + ^{- {DS}}}}} = 0}}\end{matrix} & {{Eq}.\mspace{14mu} 8}\end{matrix}$

It may be concluded that with a first order filter and a perfect model,the IMC controller 16′ (FIG. 4) will have a steady state error for astep input disturbance. To reduce or eliminate the steady state errorfor a step input disturbance, a complex filter F(s) may be employed. Butsuch a filter is difficult to implement in an actual controller used tocontrol an actual integrating process. In addition, the model thatdrives the controller still contains a numerical issue in that the valueoutput by the model will grow without bound due to the exponentialcomponent of the integrating model.

SUMMARY

According to one embodiment, a method of controlling an integratingprocess includes representing the integrating process using anon-integrating process model that approximates the behavior of theintegrating process; in response to a set point change, predicting aprocess variable of the integrating process using the non-integratingmodel as part of a parametric model-based control technique; andgenerating a control output based on the predicted process variable, thecontrol output for use in controlling the integrating process.

According to another embodiment, a method of controlling an integratingprocess using a parametric model-based control technique includesrepresenting the process using a process model; applying a filter to aninverse of the process model and predicting a process variable of theprocess in response to a set point change, the filter represented by:

$\frac{\left( {{\gamma \; s} + 1} \right)^{m}}{\left( {{ɛ\; s} + 1} \right)^{n}}$

where n is greater than m and m is greater than or equal to zero, ε is afilter time constant and γ is lead time constant of the filter, and γ isselected to minimize a steady state error for a step input disturbanceto the process by representing γ by:

$\frac{1}{m}\left( {{n\; ɛ} + \hat{D}} \right)$

where {circumflex over (D)} is an estimated dead time of the process;and generating a control output for use in controlling the process.

According to another embodiment, a method of controlling a process usinga parametric model-based control technique includes modeling a behaviorof the process with a model, the model based on at least onetime-varying parameter; establishing a model output array that containsoutput values of the model for a series of time samples that includes atime sample corresponding to an estimated dead time of the process;revising at least a portion of the model output array that correspondsto time samples occurring after the dead time of the process, therevising based on a change in the time-varying parameter; establishing adisturbance array that contains disturbance values for the process forthe series of time samples; revising at least a portion of thedisturbance array that corresponds to the revised portion of the modeloutput array, the revising conducted such that, for each revised timesample, a predicted process value determined after the change intime-varying parameter is the same as a predicted process valuedetermined before the change; and generating a bumpless transfer in acontrol output based on the revised model output array and the reviseddisturbance array in response to the change in time-varying parameter,the control output for use in controlling the process.

According to yet another embodiment, a method of controlling a processusing a parametric model-based control technique includes modeling abehavior of the process with a model, the model based on at least onetime-varying parameter; establishing a model output array that containsoutput values of the model for a series of time samples that includes atime sample corresponding to an estimated dead time of the process;revising at least a portion of the model output array that correspondsto time samples occurring after the dead time of the process, therevising based on a change in the time-varying parameter; and generatinga bumped transfer in a control output based on the revised model outputarray and a disturbance of the process in response to the change intime-varying parameter, the control output for use in controlling theprocess.

According to still another embodiment, a method of controlling a processusing a parametric model-based control technique includes modeling abehavior of the process with a model, the model including an estimate ofdead time of the process; establishing a model output array thatcontains output values of the model for a series of time samples thatincludes a time sample corresponding to the estimated dead time of theprocess; in response to a change in the estimated dead time, reindexingthe model output array with the changed estimated dead time; andgenerating a control output based on the reindexed model output array,the control output for use in controlling the process.

These and further features of the present invention will be apparentwith reference to the following description and attached drawings. Inthe description and drawings, particular embodiments of the inventionhave been disclosed in detail as being indicative of some of the ways inwhich the principles of the invention may be employed, but it isunderstood that the invention is not limited correspondingly in scope.Rather, the invention includes all changes, modifications andequivalents coming within the spirit and terms of the claims appendedhereto.

Features that are described and/or illustrated with respect to oneembodiment may be used in the same way or in a similar way in one ormore other embodiments and/or in combination with or instead of thefeatures of the other embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a model-based control system used tocontrol an integrating process or a non-integrating process;

FIGS. 2 a and 2 b respectively are a graph of a step control change anda graph of a corresponding predicted process variable for an integratingprocess resulting from the control system of FIG. 1;

FIGS. 3 and 4 are simplified versions of the model-based control systemof FIG. 1;

FIG. 5 is a block diagram of a model-based control system used tocontrol an integrating or a non-integrating process;

FIG. 6 is an exemplary graph of the steady state gain of a model used bythe controller of FIG. 5 for various values of an adjustment parameter;and

FIGS. 7 and 8 are exemplary graphs of the behavior of the control systemof FIG. 5 for respective values of the adjustment parameter.

DISCLOSURE

Representative embodiments of the present invention will now bedescribed with reference to the drawings, wherein like referencenumerals are used to refer to like elements throughout. It will beunderstood that the figures are not necessarily to scale.

Control System

With reference to FIG. 5, illustrated is a control system 20 thatincludes a parametric internal model-based control (IMC) controller 22that controls an integrating process 24. The integrating process 24 mayrelate to any controllable system, such as a material storage system(e.g., a water tank used for municipal or industrial water storage), aheating or cooling system, a furnace, a heat exchanger, a gas or liquidsupply system, a waste water treatment facility, an industrial ormanufacturing process, a process exhibiting integrating processbehavior, and so forth.

The controller 22 may be implemented with a computing device, such as asystem that has a processor for executing logical instructions and amemory for storing such logical instructions in non-volatile memory(e.g., a hard drive, an optical disk (e.g., a CD or DVD) or a flashmemory) and/or in system memory (e.g., a random access memory or RAM).The model-based control functions described herein may be implemented inexecutable code (e.g., software) that is executed by the computingdevice. In other arrangements, the controller 22 and underlyingmodel-based control functions described herein may be implemented inhardware, firmware or circuitry. Combinations of executable code,firmware and/or circuitry also may be employed. If implemented inexecutable code, the executable code may be embodied as a machinereadable medium (e.g., instructions stored by a memory device).

The controller 22 may include a model 26 (e.g., G_(m1)(s)) that isrepresentative of the process 24 and a controller function 28 (e.g.,G_(c)(s)) as will be described in greater detail below.

Control of Integrating Process

The following control techniques have application to the design of acontroller used for the parametric model-based control of an integratingprocess, such as a process represented in equations 1A (appearingabove), 1B (appearing below) or 1C (appearing below). For example, thecontroller 22 may be used to control the process 24 where the controller22 employs a process model cascade used to approximate the integratingprocess 24.

An exemplary controller design in which the techniques of thisdisclosure may be included is set forth in U.S. Pat. No. 5,191,521, thedisclosure of which is herein incorporated by reference in its entirety.This patent discloses a method of controlling a process that has atleast two process variables and a corresponding modular multivariablecontrol apparatus. When applied to the control of a multivariableprocess, the techniques described herein may be used in the control ofone or more of the process variables.

As described in the background section, the conventional model-basedcontrol technique for an integrating process is to select a model thatis an accurate representation of the process. These models arerepresented by example in equations 1A and 2.

To avoid the mathematical issue of the output of the model increasingwithout bound and to reduce the complexity of the controller 22, adifferent model and/or filter may be used to drive the control of theintegrating process 24. In one embodiment, a first order model is usedto approximate the gain and integrator (e.g., k/s) of the integratingprocess 24. An exemplary first order model that may be applied isα{circumflex over (k)}/(αs+1) where α is an adjustment factor as will bedescribed in greater detail below and is preferably selected to be arelative large number (e.g., about 100 to about 1,000). The term{circumflex over (k)} is an estimate of the steady state gain of theintegrating process and may be derived from the steady state slope of aprocess variable (PV) corresponding to the output of the process 24(e.g., a level of a storage tank or a level temperature for a thermalprocess). Applying this exemplary model to the integrating process 24,equation 1A may be rewritten as equation 9A, where {circumflex over (D)}is an estimated dead time.

$\begin{matrix}{{G_{m\; 1}(s)} = {\frac{\alpha \; \hat{k}}{{\alpha \; s} + 1}^{{- \hat{D}}S}}} & {{{Eq}.\mspace{14mu} 9}A}\end{matrix}$

The model used to approximate the integrating process 24 may be arepresentation of a non-integrating process where, for a first ordersystem, the ratio of PV(s) to control output (CO(s)) (e.g., PV(s)/CO(s))equals (G/(Ts+1))e^(−DS) where G is the gain and equals the change in PVdivided by the change in CO, T is the time constant, S is a Laplacevariable and D is the dead time.

In the approximated model represented by equation 9A, the pole is equalto −1/α. Thus, the pole is not at origin (origin being when s equalszero) as is found in equation 2 for the conventional model approach. Asthe value of α increases or is selected to be sufficiently large, thepole as defined by −1/α will approach zero and, under thesecircumstances, G_(m1)(s) as expressed by equation 9A is a satisfactoryapproximation of the perfect model G_(m)(s). Under the approximation,the initial slope of PV is {circumflex over (k)}, which corresponds tothe steady state gain of the integrating model as set forth in equation2.

As an example of the behavior of model G_(m1)(s) versus the behavior ofthe model G_(m)(s), reference is made to FIG. 6. In FIG. 6, the steadystate gain of the integrating model G_(m)(s) is represented by curve 30and curves 32 a through 32 e represent steady state gains of the modelG_(m1)(s) for various values of α. In particular, curve 32 a correspondsto an α of 1000, curve 32 b corresponds to an α of 800, curve 32 ccorresponds to an α of 500, curve 32 d corresponds to an α of 300, andcurve 32 e corresponds to an α of 100. In FIG. 6, the x-axis representstime and the y-axis represents PV. In the illustrated example, the unitsof time are in seconds, but could be other units (e.g., milliseconds,minutes or hours) depending on the process. The illustrated PV-axisincludes values to provide an indication of scale, but actual units willdepend on the process 24. In one example, the process 24 may correspondto a water storage tank such that PV is water level and the units may bein inches, feet or other distance unit.

As will be explained in greater detail below, the approximation ofequation 9A is not limited to a first order integrating process. Rather,the approximation approach may be modified for use in controlling ahigher order integrating process and/or for controlling a process withmultiple-inputs (e.g., multiple input model-based control). In addition,the approximation approach may be used in a modular multivariablecontrol arrangement, such as the process control set forth inabove-referenced U.S. Pat. No. 5,191,521.

An implementation of the controller 22 of FIG. 5 may be considered torepresent a combination of a process model (e.g., G_(m1)(s)) and arealizable model inverse (e.g., G_(c)(s)). The realizable model inversefor the controller 22 may be expressed in accordance with equation 3B.

G _(c)(s)=G _(m+) ⁻¹ ·F(s)  Eq. 3B

If a second or higher order process model is employed for theintegrating process 24, the process 24 may be represented by equation 1Bthat includes an integrator and a lag τ. Even in this case, a firstorder approximated model structure for the controller 22 may be used asrepresented by equation 9B, where the dead time estimate is modified toinclude an estimated lag {circumflex over (τ)}. Using equation 9B, asecond or higher order integrating process may be approximated with afirst order model that include the adjustment factor α and where a timeconstant portion of the process in merged into a dead time estimate.

$\begin{matrix}{{G_{p}(s)} = {\frac{k}{s\left( {{\tau \cdot s} + 1} \right)}^{- {DS}}}} & {{{Eq}.\mspace{14mu} 1}B} \\{{G_{m\; 1}(s)} = {{\frac{\alpha \; \hat{k}}{{\alpha \; s} + 1}^{{- \overset{\sim}{D}}S}\mspace{14mu} {where}\mspace{14mu} \overset{\sim}{D}} = {\hat{D} + \hat{\tau}}}} & {{{Eq}.\mspace{14mu} 9}B}\end{matrix}$

If a higher order model structure for the controller 22 is desired foruse in controlling a higher order integrating process 24, it may bepossible to approximate only the integrator portion using the expressionof equation 9A, and keep the rest of model structure the same. Asindicated, a more detailed description of higher order process models isdiscussed below.

It may be observed that the initial slope of the approximated models asset forth in equations 9A and 9B are the same as or very similar to thesteady state gain of the integrated model as set forth in equations 1Aand 1B. As a result, a corresponding transient response of the IMCcontroller 22 is achieved. For equations 9A and 9B, this result isexpressed as equation 10. Similar results for higher order formulationare achieved as expressed in equations 9C and 1C appearing in asubsequent portion of this disclosure.

$\begin{matrix}{{s\overset{\lim}{\rightarrow}{\infty \mspace{11mu} \frac{}{s}{G_{{m\; 1} +}(s)}}} = {{s^{2}\frac{\alpha \; \hat{k}}{{\alpha \; s} + 1}\frac{1}{s}} = {\frac{\alpha \; \hat{k}}{\alpha} = \hat{k}}}} & {{Eq}.\mspace{14mu} 10}\end{matrix}$

Continuing to consider the case of a first order integrating process(e.g., as expressed by equation 1A), one may select a first orderfilter, such as F(s) equals 1/(εS+1). In this case, equations 11 and 12may be derived.

$\begin{matrix}{G_{c} = {{G_{{m\; 1} +}^{- 1} \cdot {F(s)}} = {\frac{{\alpha \; s} + 1}{\alpha \; \hat{k}}\frac{1}{{ɛ\; s} + 1}}}} & {{Eq}.\mspace{14mu} 11} \\{G_{IMC} = {\frac{G_{c}}{1 - {G_{c}G_{m\; 1}}} = {\frac{\frac{{\alpha \; s} + 1}{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1} \right)}}}{1 - \frac{^{{- \hat{D}}S}}{{\alpha \; s} + 1}} = \frac{{\alpha \; s} + 1}{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}}}}} & {{Eq}.\mspace{14mu} 12}\end{matrix}$

One may represent a step input as r, such that SP(s) equals (1/s)r andequation 13A then follows from equation 5A. Equation 13A indicates thatif no disturbance is present, the PV may equal the set point.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; {\frac{s\left( {G_{IMC}G_{p}} \right)}{1 + {G_{IMC}G_{p}}} \cdot \frac{1}{s} \cdot r}}}} \\{= {s\overset{\lim}{\rightarrow}{0\mspace{11mu} {\frac{\frac{{\alpha \; s} + 1}{\alpha \; \hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}\frac{k}{s}^{- {DS}}}{1 + {\frac{{\alpha \; s} + 1}{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}}\frac{k}{s}^{- {DS}}}} \cdot r}}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} {\frac{\left( {{\alpha \; s} + 1} \right)k\; ^{- {DS}}}{\begin{matrix}{{\alpha \; \hat{k}{s\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}} +} \\{\left( {{\alpha \; s} + 1} \right)k\; ^{- {DS}}}\end{matrix}} \cdot r}}} = r}}\end{matrix} & {{{Eq}.\mspace{14mu} 13}A}\end{matrix}$

For a step input disturbance where d₁(s) equals d₁/s, equation 13B showsthat no steady state error for a step input disturbance may result.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; \frac{G_{p}}{1 + {G_{IMC}G_{p}}}d_{1}}}} \\{= {s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{\frac{k}{s}^{- {DS}}}{1 + {{\frac{{\alpha \; s} + 1}{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}} \cdot \frac{k}{s}}^{- {DS}}}}d_{1}}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{\alpha \; {{\hat{k}\left( {{\alpha \; s} + 1 - ^{{- \hat{D}}S}} \right)} \cdot k}\; ^{- {DS}}}{\begin{matrix}\left\lbrack {{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}s} +} \right. \\\left. {\left( {{\alpha \; s} + 1} \right)k\; ^{- {DS}}} \right\rbrack\end{matrix}}d_{1}}} = 0}}\end{matrix} & {{{Eq}.\mspace{14mu} 13}B}\end{matrix}$

Similarly, for a step output disturbance where d₂(s) equals d₂/s,equation 13C shows that no steady state error for a step outputdisturbance may result.

$\begin{matrix}\begin{matrix}{{t\overset{\lim}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\lim}{\rightarrow}{0\; \frac{1}{1 + {G_{IMC}G_{p}}}d_{2}}}} \\{= {{s\overset{\lim}{\rightarrow}{0\mspace{11mu} \frac{\alpha \; {{\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)} \cdot s}}{\begin{matrix}{{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}S}} \right)}} +} \\{\left( {{\alpha \; s} + 1} \right)k\; ^{- {DS}}}\end{matrix}}d_{2}}} = {{\frac{0}{k}d_{2}} = 0}}}\end{matrix} & {{{Eq}.\mspace{14mu} 13}C}\end{matrix}$

As will be appreciated by one of ordinary skill in the art, a filter(e.g., the above-noted filter F(s)) that is relatively simple toimplement may be used in conjunction with the approximated model. Inaddition, the approximated model avoids use of an integrator such thatthe possibility of values growing without bound are kept to a minimum.As will be described in greater detail below, similar results may beachieved using higher order approximated models in the structure of thecontroller 22.

With continuing reference to FIG. 5, computational error of thecontroller 22, if any, may be represented by disturbance d₃. Thecomputational error is most often attributed to rounding of numbers andis sometimes referred to as a rounding error. An exemplary term forexpressing how the disturbance d₃ impacts the calculation of predictedPV is found in equation 14A.

$\begin{matrix}{{{PV}(s)} = {\frac{\frac{1}{1 - {G_{C}G_{m}}}G_{p}}{1 + {G_{IMC}G_{p}}}d_{3}}} & \text{Eq.~~14A}\end{matrix}$

Even though computational error of the controller 22 may be very smallin value, under an integrating model (e.g., the model of equation 2), itis still theoretically possible that numerical instability may result asthe value of the computational error may grow without bound. However,using the approximated model described herein, it may be shown throughequation 14B that the disturbance d₃ may be translated to PV as arounding error magnified by α{circumflex over (k)} such that thedisturbance has minimal or no numerical instability.

$\begin{matrix}{{t\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{\infty \; {{PV}(t)}}} = {{s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}\; {0\left( {\frac{\frac{1}{1 - {G_{C}G_{m\; 1}}}G_{p}}{1 + {G_{IMC}G_{p}}}d_{3}} \right)}} = {\alpha \; {\hat{k} \cdot d_{3}}}}} & \text{Eq.~~14B}\end{matrix}$

Tuning

The foregoing model-based control of an integrating process may beimplemented in a relatively simple manner. In addition, steady stateerrors for a step set point change, input disturbances and outputdisturbances are minimized or eliminated. The model also has a highdegree of numerical stability in that it is predicted that the outputvalue of the model will not grow without bound.

To gain the most control over the integrating process 24, it may bedesirable to tune the IMC controller 22. One tunable parameter is thefilter time constant ε. Tuning the filter time constant may be used toadjust the trajectory (e.g., rate of change) of the output value and maybe accomplished in any conventional manner.

Another tuning parameter is the selection of the value for α, as thisvalue has an impact on the behavior of the control system 20 asdemonstrated in connection with FIG. 6 and will be described in greaterdetail below. The tuning of a will first be described by rearrangingequation 13B as equation 15.

$\begin{matrix}{{{PV}(s)} = {\frac{\alpha \; {\hat{k}\left( {{ɛ\; s} + 1 - ^{{- \hat{D}}\; S}} \right)}^{- {DS}}}{\left\lbrack {{\frac{\hat{k}}{k}\alpha \; ɛ\; s^{2}} + {{\alpha \left( {\frac{\hat{k}}{k} + {\left( {1 - \frac{\hat{k}}{k}} \right)^{{- \hat{D}}\; S}}} \right)}s} + ^{- {DS}}} \right\rbrack}d_{1}}} & \text{Eq.~~15}\end{matrix}$

The denominator of equation 15 indicates that the effective timeconstant is a function of α and, as α increases, the response of PV tothe set point change, the input disturbance d₁, and the outputdisturbance d₂ will be affected. In one embodiment, the value of α maybe selected to be a value of about 100 to about 1,000, but the value ofα need not be limited to this range depending on the process 24.

If a larger α is selected, the control output (CO) and the processvariable (PV) response is more stable, but the value of PV may tend toovershoot and stay slightly above the set point for a relatively longperiod of time (e.g., more than three times the selected time constant)before PV converges to the set point.

The behavior of the controller 22 for a relatively large α isgraphically illustrated in FIG. 7 where curve 34 represents a stepincrease in the set point value and curve 36 represents the PV response.In FIG. 7, α is selected to be 1,000. The x-axis in FIG. 7 representstime and the y-axis represents PV. The illustrated PV-axis includesvalues to provide an indication of scale, but the actual units willdepend on the process 24.

If a smaller α is selected, the closed-loop response is generally moreoscillatory, but PV may converge to the set point more rapidly.

The behavior of the controller 22 for a relatively small α isgraphically illustrated in FIG. 8 where curve 38 represents a stepincrease in the set point value and curve 40 represents the PV response.In FIG. 8, α is selected to be 100. The x-axis in FIG. 8 represents timeand the y-axis represents PV. The illustrated PV-axis includes values toprovide an indication of scale, but the actual units will depend on theprocess 24.

As will be appreciated, one may tune the controller 22 by selecting α tocorrespond to a desired response. Thus, α may be considered to be atuning or adjustment parameter. For instance, if response stability isof greater interest than set point tracking, a relatively large valuefor α may be selected. If faster set point tracking is desired, arelatively small α may be selected with recognition of a possibilitythat the response may have an oscillatory component.

Model-Based Control of a Time-Varying System

Time-varying systems are encountered often in various process controlapplications. In time-varying systems, the values of model gain k, modellag time constant T and model dead time D from equation 17 (appearingbelow) may change over time and/or may have an interrelationship as afunction of the behavior of the process. For example, in a water supplysystem or in a wastewater treatment system, the time for water orwastewater to flow from a first point (e.g., point A) to a second point(e.g., point B) may depend on the flow rate. If the flow rate changes,the transportation delay time (or dead time) also changes. For example,if chlorine is injected into a flow of water at point A, there is a timedelay until the chlorine is detected by a sensor downstream at point B.The time to sense the change in chlorine level may be based, in part onthe water flow rate at the time of adding the chlorine. A similarsituation arises in a production and/or packaging line wheretransportation delay time may change with conveying belt speed changes.The gain of the system may change as well. For instance, the gain of athermal process may be different at 300° F. than at 500° F.

Parameters of the system also may change as a result of programmed oruser-specified changes in model parameters to improve the accuracy ofthe model and/or to attain a desired behavior of the process. Forinstance, the selection of α as a tuning parameter may be considered achange in both gain and time constant. As such, it will be understoodthat a time-varying parameter refers to a parameter that changes nomatter the cause (e.g., process-related cause or intervention-basedcause). It will be further appreciated that most model-based controlsystems may be run in a manual mode or an automatic mode. The controlover time-varying parameters described herein has application in eithermode.

One exemplary approach to dealing with a time-varying system is to usean adaptive control technique such as a self-tuning regulator in whichthe system dynamics are identified from time to time. But adaptivecontrol has drawbacks since it is slow to adapt to changes and there arecontradicting data requirements for model identification and closed-loopcontrol. For instance, model identification under an adaptive controltechnique involves persistent excitation criteria based on frequentchanges or disturbances to the process whereas there is a preference forsmooth and stable control outputs for closed-loop control.

Another exemplary approach to dealing with a time-varying system is toapply a gain scheduling technique to the model-based control, similar togain scheduling of a proportional-integral-derivative (PID) control.This technique involves plural model predictive controls running inparallel where each control has a different model. Depending on thestate of the process (e.g., the current k, T and D), one of thecontroller outputs will be used. This technique involves the feedback ofcontroller output to all controllers, in the same fashion as an overridePID control. The gain scheduling of model based control under thistechnique is complicated to implement.

A different approach to time-varying control is presented in thisdisclosure section. For the parametric model-based controller 22, thecomputations for model prediction and controller calculation may becarried out in discrete equations, regardless of the order of the model.Exemplary discrete equations are set forth in equation 16 that representan auto-regressive moving average model. In equation 16 and ensuingequations, the y(t) terms are an array of model output values and theu(t) terms are an array of controller output values, a is thecoefficient of an auto-regressive portion of a polynomial, and b is thecoefficient of a moving average portion of a polynomial. The value t isan integer time sample such that y(t) is the model output value at thepresent time, y(t+1) is the model output value in one time sample and soforth. The value q is the number of time samples corresponding to thepoint in the array where the dead time falls.

y(t)+a ₁ *y(t−1)+a ₂ *y(t−2)+ . . . =b ₀ *u(t−q)+b ₁ *u(t−q−1)+ . ..  Eq. 16

Accordingly, for a first order system y(t+1) will equal a₁y(t)+b₀u(t)and, for a second order system, y(t+1) will equala₁y(t)+a₂y(t−1)+b₀u(t). A third term may be added for a third ordersystem and so forth.

There may be different ways of deriving the parameters a₀, a₂, . . . ,b₀, b₁, . . . for an integrating or non-integrating transfer function(e.g., transfer function G_(m)(s) or G_(m1)(s)). For example, for afirst order model cascade with a dead time given by equation 17, thecomputations may be implemented using the discrete equation given byequation 18, where a₁, is given by equation 18A, b₀ is given by equation18B and q is given by equation 18C in which Δt is a duration of thesample time t.

$\begin{matrix}{{G(s)} = {\frac{k}{{Ts} + 1}^{- {DS}}}} & \text{Eq.~~17} \\{{y(t)} = {{a_{1}{y\left( {t - 1} \right)}} + {b_{0}*{u\left( {t - q} \right)}}}} & \text{Eq.~~18} \\{a_{1} = \frac{T}{T + {\Delta \; t}}} & \text{Eq.~~18A} \\{b_{0} = \frac{{K \cdot \Delta}\; t}{T + {\Delta \; t}}} & \text{Eq.~~18B} \\{q = \frac{D}{\Delta \; t}} & \text{Eq.~~18C}\end{matrix}$

Thus, in order to predict into the future, equation 19A and 19B may beemployed, where n is the number of samples that are predicted into thefuture (e.g., n is the size of the arrays) and represents the amount offuture time that one is interested in controlling (e.g., y(t+n) at acertain desired value or set point). At current time t, the futurepredicted model output up to the dead time (e.g., y(t+1), . . . y(t+q))may be calculated using past control outputs (e.g., u(t−q+1), . . .u(t)) in accordance with equation 19A. Calculation of future predictedmodel output after the dead time (e.g., y(t+q+1), . . . y(t+n)) may bebased on future control outputs (e.g., u(t+1), . . . u(t+n−q)). Butfuture control outputs may not be known at the present time. Thus, anassumption that the control outputs will remain the same in the relativenear future may be made. Under this assumption, the future model outputsafter the dead time may be calculated in accordance with equation 19B.

$\begin{matrix}\begin{matrix}{{y\left( {t + 1} \right)} = {{a_{1}{y(t)}} + {b_{0}*{u\left( {t - q + 1} \right)}}}} \\{\vdots} \\{{y\left( {t + q} \right)} = {{a_{1}{y\left( {t + q - 1} \right)}} + {b_{0}*{u(t)}}}}\end{matrix} & {{{Eq}.\mspace{14mu} 19}A} \\{\begin{matrix}{{y\left( {t + q + 1} \right)} = {{a_{1}{y\left( {t + q} \right)}} + {b_{0}*{u\left( {t + 1} \right)}}}} \\{\vdots} \\{{y\left( {t + n} \right)} = {{a_{1}{y\left( {t + n - 1} \right)}} + {b_{0}*{u\left( {t + n - q} \right)}}}}\end{matrix}{{{where}\mspace{14mu} {u\left( {t + 1} \right)}} = {{u\left( {t + 2} \right)} = {\ldots = {{u\left( {t + n - q} \right)} = {u(t)}}}}}} & {{{Eq}.\mspace{14mu} 19}B}\end{matrix}$

In the controller 22 implementation, a dead time array (also referred toherein as a model output array) of n elements may be used for storingmodel output given by equation 20 in which n is greater than q.

y(t+n), y(t+n−1), . . . y(t+q+1), y(t+q), . . . y(t+1)  Eq. 20

This dead time array may be updated at each sample time and when a newcontrol output is calculated. Since information regarding futuredisturbances is lacking, a current disturbance d(t) may be used for allfuture disturbances under the assumption that the disturbance shouldremain constant or nearly constant over the near future.

Equation 21 provides an array of disturbance values that represents thedifference between a measured process variable and a calculated modeloutput at the current time.

d(t+n)=d(t+n−1)= . . . =d(t+1)=d(t), where d(t)=PV(t)−y(t)  Eq. 21

The predicted process variable in future time may be equal to the sum ofthe model output y(t+1) and the future disturbance estimate d(t+1), suchthat the predicted PV at future time i is given by equation 22, where iis a selected number of time samples.

PV(t+i)=y(t+i)+d(t)  Eq. 22

Then, the control output may be calculated by minimizing a costsfunction of future errors in accordance with equation 23.

$\begin{matrix}{\min {\sum\limits_{i = 1}^{n - q - 1}\left( {{{PV}\left( {q + i} \right)} - {SP}} \right)^{2}}} & \text{Eq.~~23}\end{matrix}$

In some systems, when the model dynamic changes (e.g., change of k, Tand/or D), it may not be sufficient to recalculate discrete modelparameters a₁, a₂, b₀, b₁, . . . , and/or to reinitialize the dead timearray y(t+n), . . . y(t+1) and the disturbance estimate d(t). This isbecause these actions will lose model prediction information from thepast, thereby introducing disturbances into the system.

To implement a strategy to account for time-varying changes with themodel-based control framework described herein, two arrays, togetherwith the dead time array of equation 20, may be used. The first array isgiven by equation 24 and is used to store (e.g., buffer) past controloutput values. The second array is given by equation 25 and is used tostore (e.g., buffer) the current disturbance estimate d(t) under anassumption that the disturbance remains constant over the futurecalculated time period.

u(t), u(t−1), . . . u(t−q+1), u(t−q), . . . u(t−n), u(t−n+1)  Eq. 24

d(t+n), d(t+n−1), . . . d(t+q+1), d(t+q), . . . d(t+1)  Eq. 25

For changes of the model gain (e.g., a change of k to k′) and/or themodel time constant (e.g., a change of T to T′), one may recalculate themodel parameters (e.g., a₁ to a₁′ and b₀ to b₀′ and similar for higherorder systems). In the case of a first order model according to equation18, equations 26A and 26B result. Equations 26A and 26B may be used toascertain parameter values that may be applied in the discreterepresentation of the parametric model-based control (e.g., equation16), and represent updated model parameters.

$\begin{matrix}{a_{1}^{\prime} = \frac{T^{\prime}}{T^{\prime} + {\Delta \; t}}} & \text{Eq.~~26A} \\{b_{0}^{\prime} = \frac{{k^{\prime} \cdot \Delta}\; t}{T^{\prime} + {\Delta \; t}}} & \text{Eq.~~26B}\end{matrix}$

One of ordinary skill in the art will appreciate that the values of aand b may be represented in other manners, both for equations 18A and18B and for equations 26A and 26B. As one alternative example, a₁ may beexpressed as (T−Δt)/T and a₁′ may be expressed as (T′−Δt)/T′.

The model output array (equation 20) may be reconstructed to achieve abumpless transfer or a non-bumpless transfer, as is desired for theprocess 24. Implementing details of both approaches are described below.In both approaches, the controller 22 may be run in an automated mode(e.g., a change to a manual mode may be avoided). The generation of anupdated model output array y(t) and an updated disturbance array d(t)for both approaches are discussed below. The updated arrays may be usedto generate appropriate control output values using any suitablemodel-based control technique. The handling of time-varying parameters,such as gain changes and/or time constant changes, in this manner may beapplied in control systems that employ a non-integrating model of firstand higher orders and in control systems that employ an integratingmodel of first and higher orders.

Model-Based Control of a Time-Varying System—Bumpless Transfer

In model-based control of a system, the controller predicts a futureprocess variable using a model of the process, and establishes a controloutput that is input to the process so that the actual process variablein the future will be at a desired value corresponding to a set point.When a model parameter changes, the control output calculation isaffected. But for some processes, one may not want to have aninstantaneous change in control output value in response to the changein model parameter. In this situation, a bumpless transfer is typicallypreferred over a bumped transfer.

For a bumpless transfer, the model output array y(t) may be updatedusing a₁′ and b₀′ to generate a revised model output array (e.g.,y′(t+n), y′(t+n−1), . . . y′(t+q+1), y′(t+q), . . . y′(t+1)). That is,the y(t) values may be recalculated by solving for the various y valuesfrom equations 19A and 19B while using the new model parametersaccording to equations 26A and 26B. In the terms of equation 19, thepast history of control outputs (e.g., u(t), u(t−1), . . . u(t−q+1),u(t−q), . . . u(t−n)) are used and the current control output u(t) isused for future control output values (e.g., it is assumed that futurecontrol output remains constant). It is noted that some past modeloutput values (e.g., y(t), y(t−1), . . . ) may be maintained for use insecond or higher order models.

Then, the disturbance array d(t) may be updated to generate a reviseddisturbance array (e.g., d′(t+n), d′(t+n−1), . . . d′(t+q+1), d′(t+q), .. . d′(t+1)) such that d′(i)+y′(i) equals y(i)+d(1) is satisfied (e.g.,the predicted future process variable remains the same as before thechange in model parameters (a and b) over i equals 1 to n.) Thepredicted future process variable may be the sum of the model outputarray and the disturbance array (e.g., PV(t+i)=y(t+i)+d(t+i) andPV′(t+i)=y′(t+i)+d′(t+i)).

That is, using the current disturbance value d(1), the sum of thecurrent disturbance value d(1) and the previously predicted model outputvalue y(i) for each time sample and the sum of the revised disturbancevalue d′(i) and the revised model output value y′(i) for eachcorresponding time sample will result in the same predicted processvalue. In other words, the disturbance value is updated for each timesample so that there is no change in predicted process variable frombefore the revision to the terms a and b to after revision of the termsto a′and b′. As an example, if before the change in model gain k, modelag time constant T or dead time D, the predicted process value as givenby y(i)+d(1) is 110 (e.g., y(i) is 100 and d(1) is 10) and after thechange the revised y′(i) is 98, then the revised d′(i) would be 12 sothat d′(i)+y′(i) equals y(i)+d(1) is satisfied and the predicted futureprocess variable remains the same as before the change in modelparameters (a and b) over i equals 1 to n. Future disturbances may beset as d′(t+n) with no updates of the disturbance for n samples.

In another embodiment, the model output array (e.g., y(i)) and thedisturbance array (e.g., d(i)) may be updated in the manner describedabove to generate a revised model output array y′(i) and a reviseddisturbance array d′(i) for only the portion of the arrays after thedead time (e.g., revise the arrays for i equals q+1 to i equals n). Therest of both the arrays before the dead time (e.g., y(t+q) . . . y(t+1)and d(t+q) . . . d(t+1)) may be left unchanged (e.g., the updated modelparameters from equations 26A and 26B are not used for the unchangedportions of the arrays). In this embodiment, the future disturbanceestimate may be fixed as d′(t+n) for q samples. Calculations to derivethe disturbance estimate may resume after time n.

Model-Based Control of a Time-Varying System—Bumped Transfer

For some applications an instantaneous change in control output inresponse to a change in model parameters may be acceptable. In thiscase, a bumped transfer may be acceptable.

For a bumped transfer, the model output array y(t) may be updated usinga₁′ and b₀′ to generate a revised model output array (e.g., y′(t+n),y′(t+n−1), . . . y′(t+q+1), y′(t+q), . . . y′(t+1)). That is, the y(t)values may be recalculated by solving for the various y values fromequations 19A and 19B while using the new model parameters according toequations 26A and 26B. In the terms of equation 19, the past history ofcontrol outputs (e.g., u(t), u(t−1), . . . u(t−q+1), u(t−q), . . .u(t−n)) are used and the current control output u(t) is used for futurecontrol output values (e.g., it is assumed that future control outputremains constant). It is noted that some past model output values (e.g.,y(t), y(t−1), . . . ) may be maintained for use in second or higherorder models.

For the bumpless transfer described above, the entire model output arrayand the entire disturbance array are updated, or just the portions ofboth arrays after the dead time are updated. For the bumped transfer theentire model output may be revised but only a portion of the disturbancearray is revised as follows. After revision of the model output array, aportion of the disturbance array before the dead time (e.g., d(t+q) . .. d(t+1)) is updated such that the predicted future process variable(e.g., sum of the model output array y(t+i) and the disturbance arrayd(t+i)) remains the same as before the change in model parameters for iequals 1 to q. This is the same revision as found in the bumpless case,but for only a portion of the disturbance array. The portion of thedisturbance array after the dead time (e.g., d(t+n) . . . d(t+q+1)) isleft unrevised. In this manner, the controller output may be adjusted incorrespondence to the model change. It is preferred that the disturbancearray estimate is kept constant by not calculating PV(t) minus y(t) foreach sample time for q samples and then resuming disturbance estimatecalculation at time t+q+1.

In another embodiment, a portion of the model output array y(i) isupdated only after the dead time (e.g., revise y(t) to y′(t), for iequals q+1 to i equals n) in the above-described manner. The remainderof the model output array before the dead time (e.g., y(t+q) . . .y(t+1)) and the entire disturbance array (e.g., d(t+n), . . . d(t+q+1),d(t+q), . . . d(t+1)) may be left unrevised.

Model-Based Control of a Time-Varying System—Change of Dead Time

For changes in dead time (e.g., transportation delay, thermal responsedelay, velocity change delay, etc.), one may express the delay as achange from a current dead time D to a future dead time D′. In thiscase, values of q and p may be respectively expressed in equations 27Aand 27B.

$\begin{matrix}{q = \frac{D}{\Delta \; t}} & \text{Eq.~~27A} \\{p = \frac{D^{\prime}}{\Delta \; t}} & \text{Eq.~~27B}\end{matrix}$

If the size of the dead time array (model output array) is sufficientlylarge (e.g., n is greater than q and n is greater than p), then theremay be no need to recalculate the dead time array. Rather, the dead timearray is merely relabeled (e.g., reindexed) with the new dead time p,instead of the old dead time q as set forth by equation 28.

change: y(t+n), y(t+n−1), . . . y(t+q+1), y(t+q), . . . y(t+1)

to: y(t+n), y(t+n−1), . . . y(t+p+1), y(t+p), . . . y(t+1)  Eq. 28

If n is less than p, the array may be extended, using equations 19A and19B, so that there are at least p samples in the array.

The handling of time-varying model parameters for dead time changes inthis manner may be applied to non-integrating models of first and higherorders, integrating models of first and higher orders, and may beimplemented when the controller 22 is placed in a manual mode as well asin an automatic mode.

Generalization of Model-Based Control

As indicated above, the approaches to approximating the behavior of anintegrating process with a non-integrating model may be applied to firstorder processes as well as second or higher order processes withselection of an appropriate filter.

With continued reference to FIG. 5, the following outlines thegeneralization of model-based control of an integrating process where anapproximating model of the integrating process is used in the IMCcontroller 22. Equation 5A may be generalized as represented by equation5B.

$\begin{matrix}{{{{PV}(s)} = {{{H_{O}(s)} \cdot {{SP}(s)}} + {{H_{1}(s)} \cdot {d_{1}(s)}} + {{H_{2}(s)} \cdot {d_{2}(s)}}}}{where}{{H_{O}(s)} = \frac{1}{1 + {{G_{IMC}^{- 1}(s)} \cdot {G_{p}^{- 1}(s)}}}}{{H_{1}(s)} = {\frac{G_{p}(s)}{1 + {{G_{IMC}^{- 1}(s)} \cdot {G_{p}^{- 1}(s)}}} = {{{H_{O}(s)} \cdot {G_{IMC}^{- 1}(s)}}\mspace{14mu} {and}}}}{{H_{2}(s)} = {\frac{1}{1 + {{G_{IMC}^{- 1}(s)} \cdot {G_{p}^{- 1}(s)}}} = {{H_{1}(s)} \cdot {G_{p}^{- 1}(s)}}}}} & \text{Eq.~~5B}\end{matrix}$

Also, equation 1A may be generalized by including the term G_(p) ⁰(s) asexpressed by equation 1C, where G_(p) ⁰(s) represents higher orderterms.

$\begin{matrix}{{G_{p}(s)} = {{{\frac{k}{s}{G_{p}^{0}(s)}^{- {DS}}\mspace{14mu} {where}\mspace{14mu} s}\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{14mu} {G_{p}^{0}(s)}}} = 1}} & \text{Eq.~~1C}\end{matrix}$

Using a model of the form similar to the model expressed in equation 9A,a generalized approximation model may be expressed by equation 9C, whereG_(m1) ⁰(s) represents higher order terms of the model.

$\begin{matrix}{{{G_{m\; 1}(s)} = {\frac{\alpha \; \hat{k}}{{\alpha \; s} + 1}{G_{m\; 1}^{0}(s)}^{{- \hat{D}}S}\mspace{14mu} {where}}}{{s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{14mu} {G_{m\; 1}^{0}(s)}}} = 1}} & \text{Eq.~~9C}\end{matrix}$

As will be understood, the model of equation 9C approximates anintegrating process when α is relatively large. In the limit of equation9C, α becomes infinite and G_(m1)(s) from equation 9C reduces to themodel structure found for G_(p)(s) from equation 1C.

Furthermore, the filter may be generalized as found in equation 29,where γ is a lead time constant of the filter.

$\begin{matrix}{{{G_{C}(s)} = {{G_{{m\; 1} +}^{- 1}(s)} \cdot {F(s)}}}{where}{{G_{{m\; 1} +}(s)} = {{G_{m\; 1}(s)}^{\hat{D}\; S}\mspace{14mu} {and}}}{{F(s)} = {{\frac{\left( {{\gamma \; s} + 1} \right)^{m}}{\left( {{ɛ\; s} + 1} \right)^{n}\;}\mspace{14mu} {where}\mspace{14mu} n} > m \geq 0.}}} & \text{Eq.~~29}\end{matrix}$

Using equation 4, the invertible portion of the model may be expressedas found in equation 30.

G _(IMC) ⁻¹(S)=G _(c) ⁻¹(S)−G _(m) =G _(m1+)(s)[F(s)⁻¹ −e^({circumflex over (D)}S)]  Eq. 30

If the set point SP(s) equals r/s, the PV may be expressed in accordancewith equation 31, where PV_(ss) is the steady state value of PV.

$\begin{matrix}{{PV}_{ss} = {{t\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{\infty \; {{PV}(t)}}} = {s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{11mu}\left\lbrack {s \cdot {H_{0}(s)} \cdot \frac{r}{s}} \right\rbrack}}}} & \text{Eq.~~31}\end{matrix}$

${{H_{0}(0)} = {s\overset{\lim}{\rightarrow}{0\mspace{14mu} {H_{0}(s)}}}},$

If H₀(0) is defined as then PV_(ss)=H₀(0)·r and equation 32 resultswhere θ₁ and θ₂ denote a polynomial in “s” such that

${s\overset{\lim}{\rightarrow}{0\mspace{14mu} {\theta_{1}(s)}}} = {{{0\mspace{14mu} {and}\mspace{14mu} s}\overset{\lim}{\rightarrow}{0\mspace{14mu} {\theta_{2}(s)}}} = 0.}$

Also, the solution of equation 33 is achieved and PV_(ss) equals r forall values of the indicated parameters.

$\begin{matrix}\begin{matrix}{{H_{0}(0)} = {s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{11mu}\left\lbrack \frac{1}{1 + {{{G_{{m\; 1} +}(s)}\left\lbrack {{F(s)}^{- 1} - ^{- {DS}}} \right\rbrack} \cdot \frac{s\; ^{DS}}{k\; G_{p}^{0}}}} \right\rbrack}}} \\{= \frac{1}{\begin{matrix}{{1 + s}\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\left\lbrack {\frac{\hat{k}\; \alpha \; s\; {G_{m\; 1}^{0}(s)}^{DS}}{{k\left( {{\alpha \; s} + 1} \right)}\; {G_{p}^{0}(s)}}\left\lbrack {1 + {\left( {{n\; ɛ} - {m\; \gamma}} \right)s} +} \right.} \right.}} \\\left. \left. {{s\; {\theta_{1}(s)}} - \left( {1 - {\hat{D}s} + {s\; {\theta_{2}(s)}}} \right)} \right\rbrack \right\rbrack\end{matrix}}}\end{matrix} & \text{Eq.~~32} \\{{H_{0}(0)} = {\frac{1}{s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\left\lbrack {{\frac{\hat{k}}{k}{\alpha \left\lbrack {{\left( {{n\; ɛ} - {m\; \gamma} + \hat{D}} \right)s^{2}} + {s\; {\theta_{3}(s)}}} \right\rbrack}} + 1} \right\rbrack}} = 1}} & \text{Eq.~~33}\end{matrix}$

If D₁(s) equals d₁/s, then PV_(ss) may be expressed as set forth inequation 34 for a finite value of α and if γ is selected as in equation35.

$\begin{matrix}{{PV}_{ss} = {s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\left\lbrack {{H_{O}(s)} \cdot {G_{IMC}^{- 1}(s)}} \right\rbrack}}} & \; \\\begin{matrix}{{PV}_{ss} = {{d_{1} \cdot s}\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{11mu} {G_{IMC}^{- 1}(s)}}}} \\{= {{s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{11mu} \frac{\hat{k}\; \alpha \; d_{1}}{{\alpha \; s} + 1}\left( {{n\; ɛ} - {m\; \gamma} + \hat{D}} \right)s}} = 0}}\end{matrix} & \text{Eq.~~34} \\{\gamma = {\frac{1}{m}\left( {{n\; ɛ} + \hat{D}} \right)}} & \text{Eq.~~35}\end{matrix}$

In a purely integrating model, α may equal infinity, in which casePV_(ss) equals {circumflex over (k)}d₁(nε−mγ+{circumflex over (D)}). Ifγ is selected in accordance with equation 35, PV_(ss) may equal zero.

If D₂(s) equals d₂/s, then PV_(ss) may be expressed as set forth inequation 36.

$\begin{matrix}{{{PV}_{ss} = {s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{11mu}\left\lbrack {{H_{1}(s)} \cdot {G_{p}^{- 1}(s)}} \right\rbrack}}}{{PV}_{ss} = {{s\overset{\mspace{11mu} \lim \mspace{11mu}}{\rightarrow}{0\mspace{14mu} \frac{\hat{k}\alpha}{{\alpha \; s} + 1}\left( {{n\; ɛ} - {m\; \gamma} + \hat{D}} \right){s \cdot \frac{s}{k\; {G_{p}^{o}(s)}^{- {DS}}}}}} = 0}}} & \text{Eq.~~36}\end{matrix}$

In an exemplary implementation, a filter in accordance with equation 37may be achieved by using the filter of equation 29 and selecting γ inaccordance with equation 35, selecting n to equal 1 and selecting m toequal 3.

$\begin{matrix}{{F(s)} = \frac{{\left( {{3ɛ} + D} \right)s} + 1}{\left( {{ɛ\; s} + 1} \right)^{3}}} & {{Eq}.\mspace{14mu} 37}\end{matrix}$

Using the third order filter of equation 37, G_(c) may be expressed inthe manner set forth in equations 38 and 39, which leads to theexpression of G_(IMC) as set forth in equations 40 and 41.

$\begin{matrix}{G_{c} = {{G_{{M\; 1} +}^{- 1}{F(s)}\mspace{14mu} {where}\mspace{14mu} {F(s)}} = \frac{{\left( {{3ɛ} + D} \right)s} + 1}{\left( {{ɛ\; s} + 1} \right)^{3}}}} & {{Eq}.\mspace{14mu} 38} \\{G_{c} = \frac{s\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)}{{k\left( {{ɛ\; s} + 1} \right)}^{3}}} & {{Eq}.\mspace{14mu} 39} \\{G_{IMC} = \frac{G_{c}}{1 - {G_{c}G_{m\; 1}}}} & {{Eq}.\mspace{14mu} 40} \\{G_{IMC} = {\frac{\frac{s\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)}{{k\left( {{ɛ\; s} + 1} \right)}^{3}}}{1 - {\frac{\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)}{\left( {{ɛ\; s} + 1} \right)^{3}}^{- {DS}}}} = \frac{s\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)}{k\left( {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right)}}} & {{Eq}.\mspace{14mu} 41}\end{matrix}$

Focusing on a step input disturbance term, the value of PV may beascertained through equation 42A.

$\begin{matrix}\begin{matrix}{{PV} = {\frac{G_{p}}{1 + {G_{IMC}G_{p}}}d_{1}}} \\{= \frac{\frac{k}{s}^{- {DS}}}{1 + {{\frac{s\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)}{k\left( {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right)} \cdot \frac{k}{s}}^{- {DS}}}}} \\{= \frac{\left\{ {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right\} k\; ^{- {DS}}}{s\left\{ {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}} + {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right\}}} \\{= \frac{\left( {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right)k\; ^{- {DS}}}{{s\left( {{ɛ\; s} + 1} \right)}^{3}}}\end{matrix} & {{{Eq}.\mspace{14mu} 42}A}\end{matrix}$

By differentiating both the numerator and the denominator of equation42A, the numerator portion of equation 42A may go to a value of zero insteady state as shown in equation 42B and the denominator portion ofequation 42A may go to a value of one as shown in equation 42C.

$\begin{matrix}\begin{matrix}{{numerator} = {s\overset{\lim}{\rightarrow}{0\left( {- D} \right)K\; {^{- {DS}}\left( {\left( {{ɛ\; s} + 1} \right)^{3} - {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}}} \right)}}}} \\{{+ K}\; {^{- {DS}}\left\lbrack {{3\left( {{ɛ\; s} + 1} \right)^{2}ɛ} - {\left( {- D} \right)\left( {\left( {{\left( {{3ɛ} + D} \right)s} + 1} \right)^{- {DS}}} \right.}} \right.}} \\\left. \left. {{- \left( {{3ɛ} + D} \right)}^{- {DS}}} \right) \right\rbrack \\{= {{{- {{DK}\left( {1 - 1} \right)}} + {K\left( {{3\; ɛ} + D - \left( {{3ɛ} + D} \right)} \right)}} = 0}}\end{matrix} & {{{Eq}.\mspace{14mu} 42}B} \\\begin{matrix}{{denominator} = {s\overset{\lim}{\rightarrow}{0\frac{}{s}{s\left( \left( {{ɛ\; s} + 1} \right)^{3} \right)}}}} \\{= {{s\overset{\lim}{\rightarrow}{{0\; \left( {{ɛ\; s} + 1} \right)^{3}} + {{{s\left( {{ɛ\; s} + 1} \right)}^{2} \cdot 3}ɛ}}} = 1}}\end{matrix} & {{{Eq}.\mspace{14mu} 42}C}\end{matrix}$

The third order filter, therefore, may result in PV equaling the desiredset point.

The results of this generalized approach to model-based control may besummarized as follows. The representation of an integrating processusing an approximated model may apply to second and higher orderprocesses, in addition to first order processes. Further, theapplication extends to the case where the pole at the origin isapproximated by a first order lag with a time constant that is largerelative to a dominant time constant of the process.

For an integrating process as expressed by equation 1C with anapproximated model structure as expressed by equation 9C, thesteady-state error for a step change in set point may be zero.Similarly, the steady-state error for a step input disturbance d₁ may bezero. Also, for an integrating process as expressed by equation 1C and amodel of G_(m)(s) (e.g., equation 2) that has a pole at the origin orwhen α is infinite as in equation 9C, then the steady-state error mayequal {circumflex over (k)}(nε−mγ+{circumflex over (D)})d₁. For a filterF(s) as expressed in equation 29, choosing γ to be (nε+{circumflex over(D)})/m may cause the error for a step input disturbance d₁ to againreduce to zero, although stability and transient response may depend onmodeling errors and the choice of the filter. In addition, thesteady-state error for a step output disturbance d₂ may be zero.

Focusing on the selection of filter in accordance with equation 35, thecontrol system 20 may be implemented to minimize or avoid steady stateerrors for input disturbances d₁. It is contemplated that such a filtermay be applied to various types of parametric model based control,including when an integrating model is used. This filtering approach mayhave less relevancy to applications that involve model based controlusing a non-linear optimization approach or in non-parametric types ofmodel predictive control using a step response or an impulse response.

Although the invention has been shown and described with respect tocertain preferred embodiments, it is understood that equivalents andmodifications will occur to others skilled in the art upon the readingand understanding of the specification. The present invention includesall such equivalents and modifications, and is limited only by the scopeof the following claims.

1. A method of controlling an integrating process, comprising:representing the integrating process using a non-integrating processmodel that approximates the behavior of the integrating process; inresponse to a set point change, predicting a process variable of theintegrating process using the non-integrating model as part of aparametric model-based control technique; and generating a controloutput based on the predicted process variable, the control output foruse in controlling the integrating process.
 2. The method of claim 1,wherein the control technique inverts at least an invertible portion ofthe non-integrating process model and applies a filter to the invertedportion of the model.
 3. The method of claim 2, wherein the filter isrepresented by: $\frac{1}{\left( {{ɛ\; s} + 1} \right)}$ where ε is afilter time constant and S is a Laplace variable.
 4. The method of claim2, wherein the filter is represented by:$\frac{\left( {{\gamma \; s} + 1} \right)^{m}}{\left( {{ɛ\; s} + 1} \right)^{n}}$where n is greater than m and m is greater than or equal to zero, ε is afilter time constant, γ is lead time constant of the filer and S is aLaplace variable, and where γ is selected to minimize a steady stateerror for a step input disturbance to the process by representing γ as$\frac{1}{m}\left( {{n\; ɛ} + \hat{D}} \right)$ where {circumflexover (D)} is an estimated dead time of the process.
 5. The method ofclaim 1, wherein the integrating process is a second or higher orderprocess.
 6. The method of claim 1, wherein the non-integrating processmodel is a first order model.
 7. The method of claim 6, wherein thefirst order model is represented by:$\frac{\alpha \hat{k}}{{\alpha \; s} + 1}^{{- \hat{D}}S}$ where αis an adjustment parameter, {circumflex over (k)} is an estimated steadystate gain of the process, {circumflex over (D)} is an estimated deadtime of the process, and S is a Laplace variable.
 8. The method of claim7, further comprising selecting α to tune a process variable response.9. The method of claim 8, wherein α is about 100 to about 1,000.
 10. Themethod of claim 1, wherein the integrating process is a second or higherorder process and the model is represented by$\frac{\alpha \hat{k}}{{\alpha \; s} + 1}^{{- \hat{D}}S}$ where αis an adjustment parameter, {circumflex over (k)} is an estimated steadystate gain of the process, {tilde over (D)} is a sum of an estimateddead time of the process and a lag, and S is a Laplace variable.
 11. Themethod of claim 10, further comprising selecting α to tune a processvariable response.
 12. The method of claim 11, wherein α is about 100 toabout 1,000.
 13. The method of claim 1, wherein the non-integratingprocess model is a second or higher order model.
 14. The method of claim14, wherein the model is represented by:${G_{m\; 1}(s)} = {{{\frac{\alpha \hat{k}}{{\alpha \; s} + 1}{G_{m\; 1}^{0}(s)}^{{- \hat{D}}S}\mspace{14mu} {where}\mspace{14mu} s}\overset{\lim}{\rightarrow}{0{G_{m\; 1}^{0}(s)}}} = 1}$where α is an adjustment parameter, {circumflex over (k)} is anestimated steady state gain of the process, {circumflex over (D)} is anestimated dead time of the process, S is a Laplace variable, and G_(m1)⁰(s) represents second or higher order terms of the model.
 15. A methodof controlling an integrating process using a parametric model-basedcontrol technique, comprising: representing the process using a processmodel; applying a filter to an inverse of the process model andpredicting a process variable of the process in response to a set pointchange, the filter represented by:$\frac{\left( {{\gamma \; s} + 1} \right)^{m}}{\left( {{ɛ\; s} + 1} \right)^{n}}$where n is greater than m and m is greater than or equal to zero, ε is afilter time constant and γ is lead time constant of the filter, and γ isselected to minimize a steady state error for a step input disturbanceto the process by representing γ by:$\frac{1}{m}\left( {{n\; ɛ} + \hat{D}} \right)$ where {circumflexover (D)} is an estimated dead time of the process; and generating acontrol output for use in controlling the process.
 16. A method ofcontrolling a process using a parametric model-based control technique,comprising: modeling a behavior of the process with a model, the modelbased on at least one time-varying parameter; establishing a modeloutput array that contains output values of the model for a series oftime samples that includes a time sample corresponding to an estimateddead time of the process; revising at least a portion of the modeloutput array that corresponds to time samples occurring after the deadtime of the process, the revising based on a change in the time-varyingparameter; establishing a disturbance array that contains disturbancevalues for the process for the series of time samples; revising at leasta portion of the disturbance array that corresponds to the revisedportion of the model output array, the revising conducted such that, foreach revised time sample, a predicted process value determined after thechange in time-varying parameter is the same as a predicted processvalue determined before the change; and generating a bumpless transferin a control output based on the revised model output array and therevised disturbance array in response to the change in time-varyingparameter, the control output for use in controlling the process. 17.The method of claim 16, wherein the process is a second or higher ordersystem and is represented by y(t) that equals a₁*y(t−1)+a₂*y(t−2)+ . . .+b₀*u(t−q)+b₁*u(t−q−1)+ . . . , where a is a coefficient correspondingto an autoregressive portion of a polynomial, b is a coefficientcorresponding to a moving average portion of the polynomial, u(t) is anarray of controller output values and q is a time sample indexcorresponding to the dead time.
 18. The method of claim 16, wherein themodel outputs values are represented by y(t) that equalsa₁y(t−1)+b₀*u(t−q), where a₁ is a coefficient corresponding to anautoregressive portion of a polynomial, b₀ is a coefficientcorresponding to a moving average portion of the polynomial, u(t) is anarray of controller output values and q is a time sample indexcorresponding to the dead time.
 19. The method of claim 16, wherein thechanged time-varying variable is at least one of an estimated lag timeor an estimated gain, and the revising of the model output array iscarried out by replacing the at least one of the estimated lag time orthe estimated gain with a respective changed value.
 20. The method ofclaim 16, wherein the revising of the model output array and thedisturbance array is carried out for the entirety of each array.
 21. Themethod of claim 16, wherein the changed time-varying variable is atleast one of an estimated lag time of the process or an estimated gainof the process.
 22. The method of claim 16, further comprising, inresponse to a change in the estimated dead time, reindexing the modeloutput array with the changed estimated dead time.
 23. A method ofcontrolling a process using a parametric model-based control technique,comprising: modeling a behavior of the process with a model, the modelbased on at least one time-varying parameter; establishing a modeloutput array that contains output values of the model for a series oftime samples that includes a time sample corresponding to an estimateddead time of the process; revising at least a portion of the modeloutput array that corresponds to time samples occurring after the deadtime of the process, the revising based on a change in the time-varyingparameter; and generating a bumped transfer in a control output based onthe revised model output array and a disturbance of the process inresponse to the change in time-varying parameter, the control output foruse in controlling the process.
 24. The method of claim 23, wherein: therevising of the model output array is carried out for the entirety ofthe model output array; the method further includes: establishing adisturbance array that contains disturbance values for the process forthe series of time samples; and revising a portion of the disturbancearray that corresponds to time samples occurring before the dead time ofthe process, the revising of the disturbance array conducted such that,for each revised time sample, a predicted process value determined afterthe change in time-varying parameter is the same as a predicted processvalue determined before the change; and the disturbance of the processis represented by the revised disturbance array.
 25. The method of claim23, wherein the process is a second or higher order system and isrepresented by y(t) that equals a₁*y(t−1)+a₂*y(t−2)+ . . .+b₀*u(t−q)+b₁*u(t−q−1)+ . . . , where a is a coefficient correspondingto an autoregressive portion of a polynomial, b is a coefficientcorresponding to a moving average portion of the polynomial, u(t) is anarray of controller output values and q is a time sample indexcorresponding to the dead time.
 26. The method of claim 23, wherein themodel outputs values are represented by y(t) that equalsa₁y(t−1)+b₀*u(t−q), where a₁ is a coefficient corresponding to anautoregressive portion of a polynomial, b₀ is a coefficientcorresponding to a moving average portion of the polynomial, u(t) is anarray of controller output values and q is a time sample indexcorresponding to the dead time.
 27. The method of claim 23, wherein thechanged time-varying variable is at least one of an estimated lag timeor an estimated gain, and the revising of the model output array iscarried out by replacing the at least one of the estimated lag time orthe estimated gain with a respective changed value.
 28. The method ofclaim 23, wherein the changed time-varying variable is at least one ofan estimated lag time of the process or an estimated gain of theprocess.
 29. The method of claim 23, further comprising, in response toa change in the estimated dead time, reindexing the model output arraywith the changed estimated dead time.
 30. A method of controlling aprocess using a parametric model-based control technique, comprising:modeling a behavior of the process with a model, the model including anestimate of dead time of the process; establishing a model output arraythat contains output values of the model for a series of time samplesthat includes a time sample corresponding to the estimated dead time ofthe process; in response to a change in the estimated dead time,reindexing the model output array with the changed estimated dead time;and generating a control output based on the reindexed model outputarray, the control output for use in controlling the process.